## ONTOLOGIES AND WORLDS IN CATEGORY THEORY: IMPLICATIONS FOR NEURAL SYSTEMS

### BibTeX

@MISC{_ontologiesand,

author = {},

title = {ONTOLOGIES AND WORLDS IN CATEGORY THEORY: IMPLICATIONS FOR NEURAL SYSTEMS},

year = {}

}

### OpenURL

### Abstract

ABSTRACT. We propose category theory, the mathematical theory of structure, as a vehicle for defining ontologies in an unambiguous language with analytical and constructive features. Specifically, we apply categorical logic and model theory, based upon viewing an ontology as a sub-category of a category of theories expressed in a formal logic. In addition to providing mathematical rigor, this approach has several advantages. It allows the incremental analysis of ontologies by basing them in an interconnected hierarchy of theories, with an operation on the hierarchy that expresses the formation of complex theories from simple theories that express first principles. Another operation forms abstractions expressing the shared concepts in an array of theories. The use of categorical model theory makes possible the incremental analysis of possible worlds, or instances, for the theories, and the mapping of instances of a theory to instances of its more abstract parts. We describe the theoretical approach by applying it to the semantics of neural networks.

### Citations

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Citation Context ...es. We call the theories concepts, and in the hierarchy they are organized from the abstract to the specific. They are represented incrementally in a neural network as it learns through Hebbian-like (=-=Hebb 1949-=-) synaptic potentiation (and possibly de-potentiation as well). The relationships between concepts that organize them into a hierarchy are learned along with the concepts; they express the inheritance... |

316 |
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Citation Context ...sms have the form: Morphism ri : R ! Si : Points 7! Points Lines 7! Lines Num 7! Num spatial 7! spatial on 7! on L:Lines 7! Li:Lines Notice that the only change between R and Si is the Skolemization (=-=Andrews, 1986-=-, pp. 123–127) of the existentially quantified variable n of sort Num in R by the constant i of sort Num in theory Si. This effectively declares that the line constant L in R is a spatial invariant, e... |

255 |
A survey and critique of techniques for extracting rules from trained artificial neural networks
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(Show Context)
Citation Context ...raven and Shavlik 1993; Bartfai 1994; Carpenter and Tan 1995; Mitra and Pal 1995; Pinkas 1995; Sima 1995; Kasabov 1996; Healy and Caudell 1997; Heileman et al., 1997; Healy 1999a), and the survey in (=-=Andrews et al., 1995-=-). Presumably, when a well-designed network adapts its connection weights, it is effectively modifying its responses to input stimuli to improve its future response according to some criterion. Inform... |

184 |
Basic Category Theory for Computer Scientists
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(Show Context)
Citation Context ...d in the system’s design and in its store of already-processed data. The use of a mathematical language to understand the knowledge content of a computational system is called mathematical semantics (=-=Pierce 1991-=-).ONTOLOGIES AND WORLDS IN CATEGORY THEORY 167 Cognitive neuroscientists seek to understand the relationship between structure and function in the brain – the semantics of neuron/synapse organization... |

176 |
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(Show Context)
Citation Context ...which we call knowledge coherence. Category theory, the branch of mathematics upon which our semantic theory is based, has seen increasing use in computational semantic modeling and its applications (=-=Meseguer 1989-=-; Pierce 1991; Tse 1991; Goguen and Burstall 1992; Crole 1993; Jullig and Srinivas 1993; Vickers 1993; Stoltenberg-Hansen et al., 1994; Williamson et al., 2001). We have recently become aware of other... |

153 |
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Citation Context ...cs of neuron/synapse organization. One of their significant findings is that neurons and their synaptic connections are organized on a larger scale into a system of interconnected functional regions (=-=Damasio 1989-=-; Hirsch et al., 2001). Each region is associated with one or more sensory modalities, motor control, planning, and the control of working memory (Squire and Zola-Morgan 1991; Otto et al., 1992; Rao e... |

112 |
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(Show Context)
Citation Context ...theory is based, has seen increasing use in computational semantic modeling and its applications (Meseguer 1989; Pierce 1991; Tse 1991; Goguen and Burstall 1992; Crole 1993; Jullig and Srinivas 1993; =-=Vickers 1993-=-; Stoltenberg-Hansen et al., 1994; Williamson et al., 2001). We have recently become aware of other applications of category theory to biological systems and their semantics (Rosen 1958a, b; Ehrevan a... |

111 |
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(Show Context)
Citation Context ... to analyze existing neural architectures, both biological and artificial, and can also guide the design of new artificial neural architectures. An existing computerized system for knowledge sharing (=-=Gruber and Olsen 1994-=-) uses a declarative language resembling a first-order predicate calculus to express ontologies. An ontology written in the language serves as a basis for communication among knowledgebased systems or... |

88 |
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(Show Context)
Citation Context ...s important as the theories themselves, for they give the ontology its internal coherence. The encoding of an ontology in category theory (Mac Lane 1971; Adamek et al., 1990; Pierce 1991; Crole 1993; =-=Lawvere and Schanuel 1997-=-) addresses this fact, for it has the form of a category of theories. In a category, the relationships or arrows or morphisms represent the structural imperative, the reason d’ etre that enables a cat... |

82 |
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(Show Context)
Citation Context ...f mathematics upon which our semantic theory is based, has seen increasing use in computational semantic modeling and its applications (Meseguer 1989; Pierce 1991; Tse 1991; Goguen and Burstall 1992; =-=Crole 1993-=-; Jullig and Srinivas 1993; Vickers 1993; Stoltenberg-Hansen et al., 1994; Williamson et al., 2001). We have recently become aware of other applications of category theory to biological systems and th... |

60 | Abstract and Concrete Categories, The Joy of Cats
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(Show Context)
Citation Context ...tion. The relationships between theories are as important as the theories themselves, for they give the ontology its internal coherence. The encoding of an ontology in category theory (Mac Lane 1971; =-=Adamek et al., 1990-=-; Pierce 1991; Crole 1993; Lawvere and Schanuel 1997) addresses this fact, for it has the form of a category of theories. In a category, the relationships or arrows or morphisms represent the structur... |

60 | Categories for the working mathematician. Springer-Verlag. (This is the standard reference for mathematicians, written by one of the two co-discoverors of category theory. The other was - Lane, S - 1971 |

56 | Are cortical models really bound by the "binding problem"?, Neuron 24(1
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Citation Context ...ma, 1995; Kasabov, 1996; Healy and Caudell, 1997; Heileman et al., 1997; Healy, 1999a), and discrete neural network models have been applied to biological investigations in neuroscience (Perry, 1999; =-=Riesenhuber and Poggio, 1999-=-). 6.1. Architectures An architecture A has nodes pi ði 1; 2; ...; nNÞ and connections ck ðk 1; 2; ...; nCÞ. The nodes have signal functions / i : R ! R, where R a convenient number system. For ou... |

46 | Learning symbolic rules using artificial neural networks
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(Show Context)
Citation Context ... and Pitts 1943), investigators in logic and the semantics of computation have sought an accurate understanding of the implicit meaning of neural computations; see, for example, (Arbib 1987; Fu 1992; =-=Craven and Shavlik 1993-=-; Bartfai 1994; Carpenter and Tan 1995; Mitra and Pal 1995; Pinkas 1995; Sima 1995; Kasabov 1996; Healy and Caudell 1997; Heileman et al., 1997; Healy 1999a), and the survey in (Andrews et al., 1995).... |

32 |
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Citation Context ...omputation have sought an accurate understanding of the implicit meaning of neural computations; see, for example, (Arbib 1987; Fu 1992; Craven and Shavlik 1993; Bartfai 1994; Carpenter and Tan 1995; =-=Mitra and Pal 1995-=-; Pinkas 1995; Sima 1995; Kasabov 1996; Healy and Caudell 1997; Heileman et al., 1997; Healy 1999a), and the survey in (Andrews et al., 1995). Presumably, when a well-designed network adapts its conne... |

26 | The representation of biological systems from the standpoint of the theory of categories - Rosen - 1958 |

24 | Hierarchical Clustering with ART Neural Netwoks
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(Show Context)
Citation Context ...gators in logic and the semantics of computation have sought an accurate understanding of the implicit meaning of neural computations; see, for example, (Arbib 1987; Fu 1992; Craven and Shavlik 1993; =-=Bartfai 1994-=-; Carpenter and Tan 1995; Mitra and Pal 1995; Pinkas 1995; Sima 1995; Kasabov 1996; Healy and Caudell 1997; Heileman et al., 1997; Healy 1999a), and the survey in (Andrews et al., 1995). Presumably, w... |

21 | A Relational Theory Of Biological Systems - Rosen - 1958 |

17 | Acquiring rule sets as a product of learning in a logical neural architecture
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(Show Context)
Citation Context ...plicit meaning of neural computations; see, for example, (Arbib 1987; Fu 1992; Craven and Shavlik 1993; Bartfai 1994; Carpenter and Tan 1995; Mitra and Pal 1995; Pinkas 1995; Sima 1995; Kasabov 1996; =-=Healy and Caudell 1997-=-; Heileman et al., 1997; Healy 1999a), and the survey in (Andrews et al., 1995). Presumably, when a well-designed network adapts its connection weights, it is effectively modifying its responses to in... |

15 | Specware TM : Formal Support for Composing Software - Srinivas, Jullig - 1995 |

13 | Interconnected large-scale systems for three fndamental cognitive tasks revealed by functional MRI - Hirsch, Moreno, et al. - 2001 |

10 |
Parallel distributed
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Citation Context ...nly in the most general manner. Note, however, that many artificial neural network models in the literature represent the results of neural computations at discrete steps (McCullough and Pitts, 1943; =-=McClelland et al., 1986-=-; Arbib, 1987; Fu, 1992; Craven and Shavlik, 1993; Andrews et al., 1995; Mitra and Pal, 1995; Pinkas, 1995; Sima, 1995; Kasabov, 1996; Healy and Caudell, 1997; Heileman et al., 1997; Healy, 1999a), an... |

10 |
A Unifying Framework for Structured Analysis and Design Models: An Approach using Initial Algebra Semantics and Category Theory
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(Show Context)
Citation Context ...rence. Category theory, the branch of mathematics upon which our semantic theory is based, has seen increasing use in computational semantic modeling and its applications (Meseguer 1989; Pierce 1991; =-=Tse 1991-=-; Goguen and Burstall 1992; Crole 1993; Jullig and Srinivas 1993; Vickers 1993; Stoltenberg-Hansen et al., 1994; Williamson et al., 2001). We have recently become aware of other applications of catego... |

9 | A topological semantics for rule extraction with neural networks - Healy - 1999 |

6 |
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(Show Context)
Citation Context ...iraling cost and implanted logical errors haunting software development and maintenance efforts (Jullig and166 MICHAEL JOHN HEALY AND THOMAS PRESTON CAUDELL Srinivas 1993; Srinivas and Julig 1995), (=-=Williamson and Healy 1997-=-, 2000; Healy and Williamson 2000). Another, more recent thread is perhaps closer to the spirit of the original development of category theory and its application in mathematical semantics: The explor... |

4 | Applying category theory to derive engineering software from encoded knowledge - Healy, Williamson - 2000 |

4 |
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(Show Context)
Citation Context ...al regions (Damasio 1989; Hirsch et al., 2001). Each region is associated with one or more sensory modalities, motor control, planning, and the control of working memory (Squire and Zola-Morgan 1991; =-=Otto et al., 1992-=-; Rao et al., 1997; Wickelgren 1997), and a region implicated in self-referential processing has been tentatively identified in humans (Kelley et al., 2002). One way to regard this is to assign to eac... |

3 | Caudell: 2004, Neural Networks, Knowledge, and Cognition: A Mathematical Semantic Model Based upon Category Theory - Healy, P |

3 |
Testing a Computational Account of Category-Specific Deficits
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(Show Context)
Citation Context ...kas, 1995; Sima, 1995; Kasabov, 1996; Healy and Caudell, 1997; Heileman et al., 1997; Healy, 1999a), and discrete neural network models have been applied to biological investigations in neuroscience (=-=Perry, 1999-=-; Riesenhuber and Poggio, 1999). 6.1. Architectures An architecture A has nodes pi ði 1; 2; ...; nNÞ and connections ck ðk 1; 2; ...; nCÞ. The nodes have signal functions / i : R ! R, where R a co... |

3 | Deriving engineering software from requirements - Williamson, Healy - 2000 |

2 |
Affine and Projective Geometry
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(Show Context)
Citation Context ...m labeled Two-points-define-a-line uses the predicate on with its two arguments, a point and a line, to decree that any two distinct points are ‘‘on’’ some unique line. As discussed in, for example, (=-=Bennet 1995-=-), this axiom serves as a basis for several different geometries. These include Euclidean and non-Euclidean geometries as well as geometries that do not involve continua. Each can be regarded as a spe... |

2 | Colimits in memory: category theory and neural systems - Healy - 1999 |

2 | Category Theory Applied to Neural Modeling and Graphical Representations - Healy - 2000 |

2 |
The generalization capabilities of ARTMAP
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(Show Context)
Citation Context ... computations; see, for example, (Arbib 1987; Fu 1992; Craven and Shavlik 1993; Bartfai 1994; Carpenter and Tan 1995; Mitra and Pal 1995; Pinkas 1995; Sima 1995; Kasabov 1996; Healy and Caudell 1997; =-=Heileman et al., 1997-=-; Healy 1999a), and the survey in (Andrews et al., 1995). Presumably, when a well-designed network adapts its connection weights, it is effectively modifying its responses to input stimuli to improve ... |

2 |
Adaptable neuro production systems, Neurocomputing 13
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(Show Context)
Citation Context ...ding of the implicit meaning of neural computations; see, for example, (Arbib 1987; Fu 1992; Craven and Shavlik 1993; Bartfai 1994; Carpenter and Tan 1995; Mitra and Pal 1995; Pinkas 1995; Sima 1995; =-=Kasabov 1996-=-; Healy and Caudell 1997; Heileman et al., 1997; Healy 1999a), and the survey in (Andrews et al., 1995). Presumably, when a well-designed network adapts its connection weights, it is effectively modif... |

1 |
A Neural Network for Learning Rule-Based Systems
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(Show Context)
Citation Context ...cCullough and Pitts 1943), investigators in logic and the semantics of computation have sought an accurate understanding of the implicit meaning of neural computations; see, for example, (Arbib 1987; =-=Fu 1992-=-; Craven and Shavlik 1993; Bartfai 1994; Carpenter and Tan 1995; Mitra and Pal 1995; Pinkas 1995; Sima 1995; Kasabov 1996; Healy and Caudell 1997; Heileman et al., 1997; Healy 1999a), and the survey i... |

1 | Ku¨hnberger: 2005, ‘Learning Symbolic Inferences with Neural Networks - Gust, K-U |

1 | Caudell: 2001, ‘A Categorical Semantic Analysis of ART - Healy, P |

1 | Caudell: 2002, ‘Aphasic Compressed Representations: A Functorial Semantic Design Principle for Coupled ART - Healy, P |

1 | Caudell: 2003, ‘From Categorical Semantics to Neural Network Design - Healy, P |

1 | Heatherton: 2002, ‘Finding the Self ? An Event-Related fmri Study - Kelley, Macrae, et al. |

1 |
Reasoning, Nonmonotonicity and Learning
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- 1995
(Show Context)
Citation Context ...ht an accurate understanding of the implicit meaning of neural computations; see, for example, (Arbib 1987; Fu 1992; Craven and Shavlik 1993; Bartfai 1994; Carpenter and Tan 1995; Mitra and Pal 1995; =-=Pinkas 1995-=-; Sima 1995; Kasabov 1996; Healy and Caudell 1997; Heileman et al., 1997; Healy 1999a), and the survey in (Andrews et al., 1995). Presumably, when a well-designed network adapts its connection weights... |

1 |
Getting a Grip on Working
- Wickelgren
- 1997
(Show Context)
Citation Context ...l., 2001). Each region is associated with one or more sensory modalities, motor control, planning, and the control of working memory (Squire and Zola-Morgan 1991; Otto et al., 1992; Rao et al., 1997; =-=Wickelgren 1997-=-), and a region implicated in self-referential processing has been tentatively identified in humans (Kelley et al., 2002). One way to regard this is to assign to each region a system of knowledge that... |